The present invention relates generally to filter circuits and more particularly to an improved Nyquist filter and method.
The present invention relates to filters such as the kind that can be utilized in communications systems. FIG. 1 illustrates a basic block diagram of a digital communication system that utilizes pulse amplitude modulation. In this system, a pulse generator 12 receives clock pulses and binary input data. The output of pulse generator 12 will be a digital binary stream of pulses.
The pulse stream from pulse generator 12 is applied to the digital transmitting filter 14 that shapes the pulse for output to the digital-to-analog converter 15 and transmission over channel 16. Channel 16 may be a wired or wireless channel depending upon the application. The transmitted data is received at receiving filter 18. The output of filter 18 is applied to analog-to-digital converter 20. Analog-to-digital converter 20 utilizes clock pulses that are generally recovered from the transmitted data by clock recovery circuit 22. The output binary data from analog-to-digital converter 20 is a replica of the input binary stream that was provided to pulse generator 12.
Major objectives of the design of the baseband PAM system are to choose the transmitting and receiving filters 14 and 18 to minimize the effects of noise, to eliminate or minimize inter-symbol interference (ISI) and to reduce stop band energy. Inter-symbol interference can be theoretically eliminated by properly shaping the pulses of the transmitted signal. This pulse shaping can be accomplished by causing the pulse to have a zero value at periodic intervals.
Modern embodiments of pulse shaping filters use a pair of matched filters, one for transmit and one for receive. The convolution of the transmit filter with the receive filter forms the complete pulse shaping filter. Inter-symbol interference is avoided since the combined filter impulse response reaches unity at a single point and is zero periodically at every other information point (Nyquist sampling rate). The linear superposition of pulses representing a pulse train preserves bandwidth and information content. Linear superposition of band limited pulses remains band limited and sampling the combined filter at the information rate recovers the information.
FIG. 3b shows an example of a Nyquist filter impulse response. Zeros occur at the information rate, except at one information bearing point. All Nyquist filters having the same stop band are equally bandwidth limited if the time response of the filters is allowed to go to infinity. Realizable filters, however, are truncated in time since it is not possible to have an infinitely long time function. Truncation error in the time domain causes the theoretical stop band achievable by all Nyquist filters to be violated, so that out of band energy exists in excess of the stop band frequency.
The most bandwidth efficient filter is the xe2x80x9cbrick wallxe2x80x9d filter illustrated in FIG. 3a by the box (xcex1=0). The time response of this filter is shown in FIG. 3b (xcex1=0). While bandwidth efficiency is theoretically greatest for a brick wall filter as the time response approaches infinity, truncation error causes poor performance for practical and realizable approximations to the brick wall filter.
One method of producing practical filters is to allow the stop band of Nyquist compliant filters to exceed the bandwidth of the ideal brick wall filter and smoothly transition to the stop band. A class of such filters is the raised cosine filters. In the frequency domain (FIG. 3a), the raised cosine filter smoothly approaches the frequency stop band (except for the limiting brick wall filter case). The raised cosine filter is continuous at the stop band and the first derivative is continuous. The second derivative of a raised cosine filter, however, is not continuous at the stop band.
In current embodiments of most systems, the raised cosine filter is used in its matched filter version. The transmit square root raised cosine filter, which determines the spectral bandwidth efficiency of the system, is discontinuous in the first derivative at the stop band.
The preferred embodiment of the present invention utilizes a pulse shaping filter that meets the Nyquist criteria. This filter also has the property of being continuous in the frequency domain up to and including the first derivative for the square root matched filter version. Some embodiments of the invention are in fact continuous in all derivatives for the square root version and these filters are closer to the ideal brick wall filter for the same stop band.
Nyquist filters are produced from filters in the frequency domain with a fixed frequency cutoff. As is well known in the state of the art, a fixed cutoff frequency leads to an unrealizable filter of infinite duration in the time domain. To produce a realizable filter, the ideal filter is approximated by time delaying and truncating the infinite impulse response. Truncation, however, produces unintentional out of band energy. One goal that is achieved by some embodiments of the present invention is to minimize this unintentional out of band energy after the filter is truncated.
Embodiments of the present invention provide filters that give a smaller signal ripple at the truncation length for the same theoretical stop band as raised cosine filters and therefore have better attenuation in the frequency domain. Accordingly, the preferred embodiment of the present invention has better truncation performance than the raised cosine that represents state of the art design for identical theoretical stop band.
The present invention includes embodiments that have several advantages over prior art Nyquist filters such as the raised cosine filter. For example, the filter of the preferred embodiment of the present invention reduces the effects of truncation errors, by reducing the energy remaining in the terms beyond the truncation length. This attenuation leads to lower energy levels in the stop bands. For example, one embodiment filter of the present invention has been shown to provide 10 dB improvement in the filter stop band, as compared with a comparable raised cosine filter. In other words, the out-of-band transmissions are reduced by 90%, a significant improvement.
Implementation of embodiments of the present invention in a communication system provides enhanced system performance. Since the out-of-band performance is improved, adjacent channels can be moved closer together and use less frequency guard band. This feature leads to more efficient use of the available bandwidth. This advantage will be similarly be attained for subchannels within a channel.